The h Bar

Mad Spaces

09:03

Hello. It is said that the electric Feld (gradient of the potentail) is always in 90 Degrees to the equal potential spaces.
However consider this example i just drew.
The gradient of the potential should show directly towards the charge since thats how it increases the most right?
This is not however 90 degrees!
Could someone illustrate this to me please. thanks!

JingleBells

09:20

equalp
otentia
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B. Brekke

@MadSpaces I guess in the 3-dimensional case you would get equipotential surfaces, and in the case of a single point charge you would get spheres

SunsetName

09:54

Let $x'=f(x)$ and initial condition $x(0)=x_0$ assuming that $|f(x)-f(x)|\le L|x-y|$. We have $x(t)=x_0+\displaystyle\int_0^tf(t)dt$.

And also let $x_{n+1}=x_0+\displaystyle\int_0^tf(x_n(t))dt$ . Then using Mean value theorem we get $|x_{n+1}-x_n(t)|\le L\displaystyle\int_0^t|x_{n}(t)-x_{n+1}(t)|dt$. Deduce that $\max_{t\in[0,0.5L]}|x_{n+1}-x_n(t)|\le0.5\max_{t\in[0,0.5L]}|x_{n}(t)-x_{n-1}(t)|$.

Guys I can't deduce this problem. I have tried using MVT but it only shows that $|x_{n}(t)-x_{n-1}(t)|\le L\displaystyle\int_0^t|x_{n}(t)-x_{n-1}(t)|dt$. How to deduce the inequality $\max_{t\…

Good luck guys!

Charlie

13:38

@SunsetName that doesn't seem like it relates to physics

SunsetName

@Charlie Yes it relates. But in very abstract way.

$$ \begin{align} |x_{n+1}(t)-x_n(t)| &\le\int_0^t|f(x_n(u))-f(x_{n-1}(u))|\,\mathrm{d}u\\ &\end{align}$$ is what I wanted to understand.

Charlie

I don't see how this relates to physics though, not that there aren't people in here who could solve it, but still

bolbteppa

15:02

@SunsetName
\begin{align}
|x_{n+1}(t) - x_n(t)| &= |x_0 + \int_0^{t} f(x_n(t')) dt' - x_0 - \int_0^{t} f(x_{n-1}(t'))dt'| \\
&= |\int_0^{t} [f(x_n(t')) - f(x_{n-1}(t'))]dt'| \\
&\leq \int_0^{t} |f(x_n(t')) - f(x_{n-1}(t'))|dt' \\
&\leq \int_0^{t} L |x_n(t') - x_{n-1}(t')|dt' \\
&\leq L \int_0^{t} \{ \text{max}_{t'' \in [0,t]} |x_n(t'') - x_{n-1}(t'')| \} dt' \\
&= L \{ \text{max}_{t'' \in [0,t]} |x_n(t'') - x_{n-1}(t'')| \} \int_0^{t} dt' \\
&= L t \{ \text{max}_{t'' \in [0,t]} |x_n(t'') - x_{n-1}(t'')| \}

Doesn't seem to agree with what you've written as the final answer unless you meant to include an $L^2$

2

Q: Reference on partial wave expansion in the context of QFT

Weinberg is too terse on the subject, does anyone know of good reference on how partial wave expansion is used in QFT, especially in studying resonant structures.

ⵃⴰⵎⵎⴰⴷ ⵣⵉⴷⴰⵏⵉ

What could articles provide for a beginner researcher?

bolbteppa

?

You mean 'what article could you provide for a beginning researcher?'

ⵃⴰⵎⵎⴰⴷ ⵣⵉⴷⴰⵏⵉ

I mean in my point of view they only present obtained results regardless if the reader is familiar...

with the field of research or not.

bolbteppa

Read the background material then

ⵃⴰⵎⵎⴰⴷ ⵣⵉⴷⴰⵏⵉ

@bolbteppa sometimes the paper introduces new concepts which you don't find elsewhere

bolbteppa

15:13

Physics is hard, no other way to say it

ⵃⴰⵎⵎⴰⴷ ⵣⵉⴷⴰⵏⵉ

indeed.

especially

if you didn't take your courses properly.

physics require lot of reading.

bolbteppa

Even if you did take them properly :p

ⵃⴰⵎⵎⴰⴷ ⵣⵉⴷⴰⵏⵉ

I was reading assembly language for undergrad computer science after I tried to solve some problems, I took a while to think of the difference between Physics and Computer science to say that 's a huge difference.

bolbteppa

15:31

73

Q: Did Euler make the elementary mistake √-2 √-3 = √6?

The following extract is from Tristan Needham's Visual Complex Analysis, Even in 1770 the situation was still sufficiently confused that it was possible for so great a mathematician as Euler to mistakenly argue that √-2 √-3 = √6. I found this to be a bit far fetched. A simple Google search...

history mathematics history-of-science scientific-papers logic

Charlie

16:54

So the answer is $\sqrt{-2}\sqrt{-3}=-\sqrt{6}$? Is the mistake that the mneumonic way of moving around numbers inside square roots isn't valid for imaginary numbers?

ACuriousMind

Yes. But it is no mistake because Euler used the $\sqrt{}$ symbol differently, apparently.

Charlie

Ah I see

Secret

nature.com/articles/s41598-019-40765-6

hmm...

being able to effectively rerun a state as an error checking sure is useful

B. Brekke

17:32

I'm struggling to define a disconnected diagram without introducing the notion of a diagram. I want to only use the definition of contractions, does anyone have a suggestion?

ACuriousMind

@B.Brekke It's a set of contractions of fields $\phi(x_i)$ where none of the $x_i$ are external points. You can define the notion of "external point" without reference to diagrams by just saying it's a position that is not integrated over.

B. Brekke

@ACuriousMind I will go for something like that. What confused me was how to separate the notion a single field $\phi(x_i)$ and the set of fields constituting a field operator like "g$\phi(x_i)$$\phi(x_i)$$\phi(x_i)$$\phi(x_i)$"

ACuriousMind

I don't quite get what you mean

Let's say you have $\phi^4$ theory to first order. Then you have to contract a term like $\langle \phi(x_1)\phi(x_2)\int g\phi(x_3)\phi(x_3)\phi(x_3)\phi(x_3)\mathrm{d}x_3\rangle$

The disconnected piece is when you contract the first two with each other and then the four $\phi(x_3)$ with each other

All other contractions give you at least a partially connected diagram.

B. Brekke

18:03

@ACuriousMind I feel like I have an ambiguity because I denote both $g\phi(x_3)\phi(x_3)\phi(x_3)\phi(x_3)$ and $\phi(x_3)$ a field. Well, actually, I am just using operators, as I am working on a lattice, and haven't felt the need to introduce fields. The problem stands, I denote both the first and second term as "operator".

So each of my creation or annihilation operators "belong" to a larger operator(composite), such as an interaction operator, and when I contract two operators I mean the small creation and annihilation operators, but considering connectivity I actually need to consider which operator "composite" the operators belong to

Mad Spaces

18:48

Reading this book, where it is the potential of a charged rod to be calculated. the author makes the demand that the potential at the point R where is R ist the radius of the rod is set to zero. (so an object at the surface area of the charged rod has no potentail) i do not understand this. why is not the potential maximum? or infinite?

B. Brekke

@MadSpaces The potential has no physical significance, only the gradient of the potential

Mad Spaces

So why is not set like usual the the potentail at infinity is null and not at the surface.

I do not see the meaning here of this choice.

B. Brekke

@MadSpaces A matter of preference. I haven't checked, but maybe some integration simplifies or you get a simpler form for the potential

Mad Spaces

i see why.. i integrated and if we choose that the potential to infinity is null one gets Ln of r devided by infinity.. this means the Ln 0 which is unsolvable.

does it make physical sense however that the potentail Energy at surface is zero? … i know the choice is free.. but this is not intuitive.

B. Brekke

@MadSpaces Same intution as for heights. Do you measure distance above sea level, distance below the top of Mount Everest or distance from the center of the earth. It doesn't matter, all height differences are the same.

Mad Spaces

19:01

yes i believe my brain is polluted by the idea of 0 meaning stable and not moving.. i see its legit..

user434058

19:51

Can I write $(1-\sin x)$ as $\displaystyle\prod_{k=1}^{\infty}\left(1-\frac{(-1)^{k-1}\cdot 2x}{(2k-1)\pi}\right)$?

Charlie

20:27

@FakeMod is that an identity or something you derived?

seems like quite a specific mathematical identity, you could ask in chat.stackexchange.com/rooms/36/mathematics

Mostafa

Do you know how can I make a single bibliography item non-italic? It's actually a footnote not a real citation so I don't want it to appear italic

SunsetName

21:27

@bolbteppa Great answer. This is really clear!

Thank you very much.

Did you also use holder's inequality?

SunsetName

22:37

I was inspired by Physicists like R.Feymann when I was kid. Now I feel completely different I mean they never struggled in hard or in other words the problem that exist at their time only the problems no one could ask. It is hard for great physicis to struggle in easy task like finding inequality .

May be I am referring to undergraduate problems.

Do you guys feel that your intellect is limited,Feel greatly dumb or like Just feel like there is so much to learn at 21th century?

Also do you guys struggle with simple things? Or just were born geniuses.

Charlie

23:11

What do you mean "easy task like finding inequality"?

SunsetName

@Charlie like min question lol

Charlie

It's a shame so many people think you have to be born a certain way to become a great scientist, what separates an undergraduate from a world-leading scientist is, largely, tens of thousands of hours of work and practice.

What's "min question" though?

SunsetName

@Charlie Mine not min lol Typing mistake.

Charlie

That doesn't actually make it any clearer what you meant lol

Oh wait you mean "holder's inequality"

SunsetName

@Charlie lol no before that I asked one inequality question.

Charlie

23:18

Oh I see

SunsetName

Dude in my class I see bunch of geniuses who answer fast and think far more faster than mine and meanwhile I am like as slow as slug lol

I think most of the people gain success with their natural ability than spend hours figuring an answer.

Charlie

Then they have developed skill sets that allow them to answer those questions quickly

It's just practice, the more you do something the faster you will become

SunsetName

@Charlie Really? I see them playing football all the time lol

And pretty sure they sleep

Charlie

huh?

SunsetName

they have photographic memory too

you just haven't meet this kind of brilliant monsters. I certainly believe that they are monster lol.

Charlie

23:22

The idea that you're just born smart and there's nothing you can do to change that is not one I think you'll hear many serious academics promote.

SunsetName

Kinda feels it isn't fair lol.

Charlie

Put in your time and don't be afraid to revisit the basics

SunsetName

You spend hours digging knowledge on textbook and meanwhile in other side it comes naturally to them.

@Charlie Yeah that's the only way. There is no royal way to learn physics and mathematics except for few people.

It's time to study again. Later :)

Charlie

Have fun :)

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